Basic Probability
Basic probability is a branch of mathematics that deals with the likelihood of an event occurring. It involves calculating the chances of different outcomes in a given situation. This concept is fundamental in understanding uncertainty and making informed decisions in various fields such as statistics, finance, and science.
8 Key excerpts on "Basic Probability"
eBook - ePubProbability, Statistics, and Data
A Fresh Approach Using R
- Darrin Speegle, Bryan Clair(Authors)
- 2021(Publication Date)
- Chapman and Hall/CRC(Publisher)
...2 Probability DOI: 10.1201/9781003004899-2 A primary goal of statistics is to describe the real world based on limited observations. These observations may be influenced by random factors, such as measurement error or environmental conditions. This chapter introduces probability, which is designed to describe random events. Later, we will see that the theory of probability is so powerful that we intentionally introduce randomness into experiments and studies so we can make precise statements from data. 2.1 Probability basics In order to learn about probability, we must first develop a vocabulary that we can use to discuss various aspects of it. Definition 2.1. Terminology for statistical experiments: An experiment is a process that produces an observation. An outcome is a possible observation. The set of all possible outcomes is called the sample space. An event is a subset of the sample space. A trial is a single running of an experiment. Example 2.1. Roll a die and observe the number of dots on the top face. This is an experiment, with six possible outcomes. The sample space is the set S = { 1, 2, 3, 4, 5, 6 }. The event “roll higher than 3” is the set { 4, 5, 6 }. Example 2.2. Stop a random person on the street and ask them in which month they were born. This experiment has the twelve months of the year as possible outcomes. An example of an event E might be that they were born in a summer month, E = { June, July, August }. Example 2.3. Suppose a traffic light stays red for 90 seconds each cycle. While driving you arrive at this light, and observe the amount of time that you are stopped until the light turns green. The sample space is the interval of real numbers [0,90]...
eBook - ePub- Andrew F. Hayes(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...Your subjective judgment may be that the probability is very small, small, medium, large, or perhaps very large or even certain. And we intuitively understand that if the probability is very small, we know that the event is unlikely to happen, whereas if the probability is very large, the event is fairly likely to happen. This is our everyday use and understanding of probability. From a mathematical perspective, probability isn’t much more difficult that this, but statisticians like to assign precise numbers to probabilities rather than the subjective labels that we use in our day-to-day language. Probability can be a very daunting subject, and of all topics in mathematics, probability is the one area that can befuddle even experienced mathematicians. However, with some basic principles and a little patience, many problems in probability that seem complicated on the surface are not when you dig deeper. The key to success is staying organized in your thinking. But before discussing some of these basic principles, it is first worth defining the term probability. Probability has many technical definitions, but they all boil down to essentially the same basic idea. The probability of an event Y is the proportion of times that an event Y is expected to occur out of k “trials” or opportunities for the event to occur. So if the probability of Y, denoted P (Y), is 0.30 then in 1000 “trials” or opportunities, you would expect Y to occur 300 times, because (300/1000) = 0.30. A more specific way of conceptualizing probability is to consider how many different events are possible and how many meet certain criteria of interest. Probability is then defined as Pr o b a b i l i t y = N u m b e r o f q u a l i f y in g e v e n t s N u m b e r o f p o s s i b l e e v e n t s 5.1 Neither the numerator nor the denominator of equation 5.1 can be negative, and the denominator can never be smaller than the numerator. Therefore, probabilities are scaled to be between zero and one...
eBook - ePub- Mustapha Abiodun Akinkunmi(Author)
- 2019(Publication Date)
- De Gruyter(Publisher)
...4 Basic Probability Concepts In this chapter, we will discuss some basic concepts of probability, solving the problem of probability using Venn diagrams. The axioms and rules of probability will be discussed and will be extended to conditional probabilities. Practical examples with R code will be used for illustration. 4.1 Experiment, Outcome, and Sample Space As we have mentioned before, it is important to understand the definitions of certain terms to be able to use them successfully. So, the first step in gaining an understanding of probability is to learn the terminology and the rest will be a lot simpler. 4.1.1 Experiment Experiment is a measurement process that produces quantifiable results. Some typical examples of an experiment are: the tossing of a die, tossing of a coin, playing of cards, measuring weight of students, and recording growth of plants. 4.1.2 Outcome Outcome is a single result from a measurement. Examples of outcomes are: getting a sum of 9 in the tossing of two dice, turning up of heads in the toss of a coin, selecting a spade from a deck of cards, and getting a weight above a certain threshold (say 50 kg). 4.1.3 Sample Space The sample space is the set of all possible outcomes from an experiment and is denoted with S. The sample space of tossing a die is S = 1, 2, 3, 4, 5, 6 ; the sample space of tossing a coin is denoted as S = H, T. In addition, the sample space for tossing two dice. is S = 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 4, 1, 4, 2,[--=PLGO-SEP. ARATOR=--]4, 3, 4, 4, 4, 5, 4, 6, 5, 1, 5, 2, 5, 3, 5, 4, 5, 5, 5, 6, 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 6, 6 4.2 Elementary Events Any subset of a sample set, empty set, and whole set inclusive is called an event. An elementary event is an event with a single element taken from a sample space and it is denoted as E...
eBook - ePubIntroductory Probability and Statistics
Applications for Forestry and Natural Sciences (Revised Edition)
- Robert Kozak, Antal Kozak, Christina Staudhammer, Susan Watts(Authors)
- 2019(Publication Date)
- CAB International(Publisher)
...3 Probability The Foundation of Statistics We use statistical information every day to qualify statements and to help us make decisions. For example, we may hear statements like: • There is an 80% chance of rain today. • The odds are one in 13 million that you will win the lottery. Or we may be confronted with questions like: • What is the likelihood of receiving an A on the first exam in this course? • What is the chance that the Vancouver Canucks will win the next Stanley Cup? Statistical inference, the generalization from a sample to a population, involves drawing a conclusion about a population on the basis of available, but incomplete, information. Hence, statistical inference involves a certain amount of uncertainty, and statisticians should not base decisions on statistical inference unless the risk of uncertainty can be reduced to a tolerable minimum. Problems involving ‘uncertainty’, ‘chance’, ‘likelihood’, ‘odds’ and other such factors require an understanding and application of the theory of probability. Probability is the branch of mathematics that incorporates the most important set of concepts used in the field of statistics. The purpose of this chapter is to introduce the basic theories of probability that are required to appreciate and understand many of the concepts of statistical inference. 3.1 Sample Space and Events In statistics, we define an experiment as a process that produces some data. In Chapter 1, we described an experiment to study the effects of seeding date and seedbed preparation on germination. A wood scientist could be interested in studying the effect of temperature and applied pressure on the strength properties of plywood. Experi ments such as tossing a coin, rolling a dice, or drawing a card from an ordinary (52 cards) deck of cards will also produce some data...
eBook - ePubInterpreting Statistics for Beginners
A Guide for Behavioural and Social Scientists
- Vladimir Hedrih, Andjelka Hedrih(Authors)
- 2022(Publication Date)
- Routledge(Publisher)
...It is usually expressed as a number in the range between 0 and 1, denoting the proportion of the total number of outcomes in which the event occurs. A probability of 0 means that that particular outcome never occurs as the outcome of the random event, while a probability of 1 means that that particular outcome is always the outcome of the random event in question. Formulated in this way, probability is presented as an expectation on how future events will unfold. However, probability is calculated based on past events and the expectations that future will be the same as the past was. Probability is calculated by making a large number of observations of an event we view as (sufficiently) random and counting the number of times different outcomes occurred. Then we divide the number of each particular outcome by the total number of observed outcomes (of all types) and declare that this proportion is the probability of that particular outcome. Presented mathematically, it looks like this: Probability of outcome A = number of times outcome A was observed / total number of observed outcomes A key take here is that the total number of observed outcomes needs to be large in order for it to be possible for probability to be assessed like this. This has to do with a mathematical theorem called the Law of large numbers that states that the ratios of outcomes of an observed random event will become closer to their true probabilities as the number of observed events (referred to as trials) increases. When the number of observations is small it is much more likely that our outcomes deviate more from their true probabilities. For example, when throwing dice, it is much easier to obtain 6 in 2 throws in a row than in 10 throws in a row. If we threw dice a million times, it would be practically impossible that we get a 6 every time in those million times...
eBook - ePubStatistics for the Behavioural Sciences
An Introduction to Frequentist and Bayesian Approaches
- Riccardo Russo(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...3 Introduction to probability 3.1 Why are some notions of probability useful? In the previous chapters we said nothing, explicitly, about the relevance of probability in statistics. It is now important to make clear that the use of probability is at the heart of inferential statistics. Inferential statistics, as mentioned in the first chapter, is essential for deciding whether the performances obtained in two experimental conditions are different (e.g., does classical music played in the background during learning improve memory of exam material over a condition where no music is played?). The decision process involved in inferential statistics, being this either Classical/Frequentist or Bayesian, is probabilistic in nature, thus some notions of probability are essential to understand how statistical inference works. Finally, although there are conceptual differences in the way probability is defined according to the analytic, the Frequentist, and the subjective perspectives the same formulae and rules are used to calculate the probability of the occurrence of various events. These rules, embedded in probability theory, allow optimal ways to update our views and theories on the basis of the data collected. 3.2 Some preliminary definitions and the concept of probability Before defining what probability is, some preliminary definitions are required. The first thing to point out is that an experiment is characterised by the fact that there are different potential outcomes and it is not possible to predict the specific outcome that is going to occur. For example, if a die is rolled, it is not possible to predict which of the six digits will face upward...
eBook - ePub- Ayanendranath Basu, Srabashi Basu(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
...Millions of other examples may be cited. The bottom line is that a decision maker with a probabilistic understanding of the situation is far better equipped to handle the real issues compared to one who does not have this capability. In Chapter 1 we alluded to actionable insight. It is nothing but the ability to take a justifiable decision – justification based on data collected on past business performance and the story told by the data. Uncertainty is at the root of insight – if there is no uncertainty, no insight is required. Every business will then act similarly and the outcomes will be identical; there will also be no possibility of any competitive advantage. As uncertainty is a reality, the business which is able to harness the uncertainty in the best possible manner and use it advantageously is going to be the winner. The concept of probability is deeply mathematical and a certain amount of theory cannot be avoided while introducing it. We have tried to focus on the applications rather than the proofs. The entire analytics machinery and the assimilation of data are rooted in probability. The importance of understanding probability will be clearer in subsequent chapters. Any discourse on probability must start with a basic description of set theory, which is the topic of the next section. Familiarity with set theory is vital for a comprehensive development of the notion of probability. 5.1 Basic Set Theory A set is a collection of distinct objects with some common property. The objects that make up a set are usually called the elements of the set. We will refer to sets by upper case Roman letters like A, B, C, etc. Sometimes sequences of sets will be represented by subscripted letters such as A 1, A 2, …. If the object a belongs to the set A, it is symbolically represented as a ∈ A. The number of elements in the set A will be denoted by N (A)...
- A. Hasofer, V.R. Beck, I.D. Bennetts(Authors)
- 2006(Publication Date)
- Routledge(Publisher)
...We can then state that the probability that both smoke detectors are defective is 0.05 × 0.05 = 0.0025. This result is the key to understanding the principle of redundancy in system design. Suppose we install several components to carry out the same function in some system. Suppose further that the components have independent probabilities of failure. Then the probability that all components will fail is much smaller (usually at least one order of magnitude) than the probability of failure of each component. 3.7 Random variables In order to be able to handle the events in a sample space mathematically, it is useful to attach a numerical value to each elementary event. (Recall that elementary events are events that contain just one point.) The set of numerical values of the elementary events is called a random variable, usually abbreviated as r.v. Conventionally, random variables are represented by capital letters, e.g. X. When the elementary events are themselves numbers, there is a natural way of doing this. But it is sometimes useful to attach numbers to qualitative events. For example, suppose the sample space considered is that of the sex of an occupant and consists of two possible outcomes, namely { male, female }. We could assign to the event “male” the number 0 and to the event “female” the number 1. The advantage of doing this is, for example, that if we consider n occupants and attach to occupant i the random variable X i, then the number of females among the n occupants can be represented as X 1 + X 2 + · · · + X n (or equivalently by ∑ X i). 3.7.1 Types of random variables In this work we shall consider two types of random variables: discrete and continuous. A random variable is said to be discrete if its range is a discrete (finite or countable) set of real numbers. Usually, the values are {0, 1, 2,. . . }...
